Lecture for Week 10 Secs 4 5 8 Derivative Miscellany III 1 Growth and decay problems I already discussed the theory of these problems two weeks ago so let s just do an example 2 Exercise 4 5 3 A culture starts with 500 bacteria and after 3 hours there are 8000 a Find the formula for the number after t hours b Find the number after 4 hours c When will the population reach 30 000 3 The basic assumption is that the number of new bacteria is proportional to the number already there parents So B t B 0 ekt for some constant k So according to the data 8000 500e3k or 3k ln 80 5 ln 16 or k B t 500e 1 3 t ln 16 4 1 3 ln 16 500 16 t 3 Then B 4 500 16 4 3 calculator output For the last part 30 000 500 16 300 t log16 3 5 t 3 log16 60 ln 60 3 ln 16 t 3 5 Inverse trigonometric functions There are two aspects of inverse trig functions that need to be studied the definitions especially branch choices their derivatives The most important inverse trig functions are sin 1 and tan 1 6 Both of the problems we encountered for the square root function also appear for the inverse sine 1 sin is always between 1 and 1 so there is no way to define sin 1 z if z 1 unless we go into complex numbers which we won t 2 For 1 z 1 there is more than one with z as its sine In fact there are infinitely many So we have to choose a prin7 cipal value or branch of the inverse function The standard choice is to pick so that 2 2 Please refer to the book for the graphs p 276 and p 278 in this case Recall that to get the graph of an inverse function you can plot the original function on a transparent sheet and flip it over so that the horizontal and vertical axes are interchanged 8 Thus sin sin 1 z z always but sin 1 sin is false if is not in the principal interval Recall also that sin 1 z does not mean sin z 1 that is 1 sin z although sin2 z does mean sin z 2 This notational inconsistency is unfortunate but we re stuck with it Let s not even ask what sin 2 z means Another notation for the inverse is arcsin z 9 The inverse tangent is easier see graphs p 279 because it is defined for all z and all the branches look the same have positive slope But there are still infinitely many branches and the standard choice is 2 2 Why is it here but for the inverse sine This tan 1 z is a very nice function It increases smoothly between horizontal asymptotes at 2 and 2 10 The usual technique for differentiating an implicit or inverse function yields the formulas 1 d 1 sin x 2 dx 1 x 1 d 1 tan x dx 1 x2 These are ordinary algebraic functions All trace of trig seems to have disappeared One reason inverse trig functions are important is that they help provide the antiderivatives of certain algebraic functions 11 Exercise 4 6 51 Find the derivative of g x sin 1 3x 1 and state the domains of g and g Exercise Find an antiderivative of f x 3 10 2 2 x 1 4 4x 12 g x sin 1 3x 1 3 g x p 1 3x 1 2 which could be simplified For g to be defined we need 3x 1 1 Case 1 3x 1 0 Then 3x 1 1 x 0 3x 1 0 x 31 13 Case 2 3x 1 0 Then 3x 1 1 x 23 3x 1 0 x 31 So the domain of g consists of the two intervals 31 x 0 and 2 3 which fit together to give 23 x 0 14 x 13 For g to be defined we also need 3x 1 6 0 hence the interval shrinks to 32 x 0 See the vertical tangents at the ends of the graph Fig 4 on p 278 Alternative solution of the inequality 3x 1 1 x 13 13 This clearly describes the numbers whose distance from 31 is at most 13 namely the interval 2 3 0 15 F x what is F F x 10 3 2 2 x 1 4 4x 3 1 10 2 2 2 1 x x 1 so the obvious choice is F x 3 sin 1 x 10 tan 1 x 2 16 Soon we will reach the proof that the only other antiderivatives are equal to this one plus a constant What if the two numbers inside the square root were not the same Look forward to the excitement of Chapter 8 in Math 152 17 Hyperbolic functions This topic is not in the syllabus for Math 151 at TAMU To see why it should be read my paper in College Math Journal 36 2005 381 387 It also explains why I don t talk about cot csc sec 1 etc 18 Indeterminate forms l Hospital s rule The name is pronounced Loap it ALL more or less and sometimes spelled l Ho pital In my opinion the two most important things to learn about l Hospital s rule are when not to use it what it teaches us about limits of exp and ln at infinity 19 Suppose we want to calculate the limit of f x g x as x a a may be and suppose that both f x and g x approach 0 in that limit or both approach L Hospital s rule states that f x that limit is the same as the limit of g x which may be easier to calculate Please don t confuse this formula with the limit law for a quotient or with the formula for the derivative of a quotient They are three different things 20 Here is an example of the correct use of the rule sin 5x 0 lim x 0 7x 0 5 cos 5x 5 lim x 0 7 7 However you didn t really need the rule to do this problem did you You already know that sin 5x 5x when x 0 or can appeal to lim sinu u 1 u 0 21 After studying Taylor series Chapter 10 you will know many other situations where the behavior of the functions f and g near a is obvious so l Hospital is unnecessary Many students overuse l Hospital s rule relying on it as a black box when they would learn much more and solve the problems equally fast by just taking a close look at and comparing the behavior of the numerator and denominator as x a 22 Here is an example where using the rule is absolutely wrong We know that lim cosx x x 0 because the numerator approaches 1 while the denominator approaches 0 If you incorrectly applied …
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